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July 10th, 2014, 09:40 AM  #1 
Newbie Joined: Jul 2014 From: Canada Posts: 4 Thanks: 0  relationship between derivative of function and derivative of inverse
from dx/dy = 1/y', derive d^2/dy^2 = y''/(y')^3 and d^3x/dy^3 = 3(y'')^2y'y''/(y')^5. I have no idea where to start please explain. Also can you please explain why the second and higher derivatives are d^nx/dy^n and not dx^n/dy^n? 
July 10th, 2014, 03:38 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra 
$\newcommand{\d}{\mathbb{d}}$ $$\text{From} \qquad \frac{\d x}{\d y} = \frac{1}{y^\prime} \qquad \text{derive} \qquad \frac{\d^2x}{\d y^2} \qquad \text{and} \qquad \frac{\d^3x}{\d y^3}$$ We just need to differentiate, but note that $y$ is a function of $x$, and in particular $y^\prime$ is a function of $x$. Thus to differentiate with respect to $y$ we need to use the chain rule by writing \begin{align*} \frac{\d x}{\d y} &= \frac{1}{y^\prime(x(y))} &&= {y^\prime}^{1}(x(y)) \\ \frac{\d^2x}{\d y^2} &= {y^\prime}^{2}(x(y)) y^{\prime\prime}(x(y)) \frac{\d x}{\d y} &&= {y^\prime}^{3}(x(y)) y^{\prime\prime}(x(y)) &&= \frac{y^{\prime\prime}}{{y^\prime}^{3}} \\ \frac{\d^3x}{\d y^3} &= 3{ y^{ \prime\prime } (x(y)) }{ y^\prime(x(y)) }^{4}{ y^{ \prime\prime }(x(y)) }\frac{\d x}{\d y} + { y^\prime(x(y)) }^{3}y^{ \prime\prime\prime }(x(y))\frac{\d x}{\d y} &&= 3{y^{ \prime\prime }(x(y))}^2{ y^\prime(x(y)) }^{5} + { y^\prime(x(y)) }^{4}y^{ \prime\prime\prime }(x(y)) &&= \frac{3{y^{ \prime\prime }}^2 + y^\prime y^{ \prime\prime\prime }}{ { y^\prime }^{5} } \end{align*} We write $\frac{\d^2 y}{\d x^2}$ because $\frac{\d}{\d x}$ is an operator. We are not squaring $\frac{\d y}{\d x}$, but doing the operation of differentiation twice on $y$. 
July 10th, 2014, 06:57 PM  #3 
Newbie Joined: Jul 2014 From: Canada Posts: 4 Thanks: 0 
Thank you very much!


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